Question

The sum of first m terms of A.P. is $4{m^2} - m$ . If the nth term is 107, find the value of n. Also, find the 21st term of this A.P.

Answer 1

Hint- In this question we have sum of first m terms of an A.P. so we will determine first term and common difference by putting the value of m as 1 or 2, Now to proceed further we will use the formula of nth term of A.P. as ${a_n} = a + \left( {n - 1} \right)d$ by using it we will get the value of n and again by putting the value of n as 21 in this formula we will get the required result.

Complete step by step answer:
Given that the sum of first m terms of an A.P. is $4{m^2} - m$
Now to determine the first term put the value of m as 1, because the sum of the first term is first term.
$
  {S_1} = 4{m^2} - m \\
  {S_1} = 4 \times {\left( 1 \right)^2} - 1 \\
  {S_1} = 3 \\
$
Therefore, the value of first term is 3
$a = {S_1} = 3$
Also, to determine common difference, the sum of first two terms is given by
Putting the value of m as 2 in the given sum equation
$
  {S_2} = 4{m^2} - m \\
  {S_2} = 4 \times {\left( 2 \right)^2} - 2 \\
  {S_2} = 16 - 2 \\
  {S_2} = 14 \\
$
As we know that
${S_2} = {a_1} + {a_2} = a + a + d$
Now substituting the value of ${S_2} = 14,a = 3$ in the above equation, we get
$
  {S_2} = a + a + d \\
   \Rightarrow 14 = 3 + 3 + d \\
   \Rightarrow d = 14 - 6 \\
   \Rightarrow d = 8 \\
$
It is given that ${a_n} = 107$
As we know that the nth term is given as ${a_n} = a + \left( {n - 1} \right)d$
Substituting the value of ${a_n},a{\text{ and }}d$ in the above nth term equation
$
  {a_n} = a + \left( {n - 1} \right)d \\
   \Rightarrow 107 = 3 + \left( {n - 1} \right) \times 8 \\
   \Rightarrow 107 - 3 = 8n - 8 \\
   \Rightarrow 104 + 8 = 8n \\
   \Rightarrow 8n = 112 \\
   \Rightarrow n = \dfrac{{112}}{8} \\
   \Rightarrow n = 14 \\
$
Now to find the value of ${a_{21}}$
Here the value of n is 21
Using the nth term formula and substituting the value of a, n and d we get
$
  {a_n} = a + \left( {n - 1} \right)d \\
  {a_{21}} = 3 + \left( {21 - 1} \right) \times 8 \\
  {a_{21}} = 3 + \left( {20} \right) \times 8 \\
  {a_{21}} = 3 + 160 \\
  {a_{21}} = 163 \\
$
Hence, the value of 21 terms is $163$ and the value of n is 14.

Note- In order to solve these types of questions first of all remember the formula of the nth term of A.P. Also remember the formula of sum of n terms of an A.P. In this question the sum of m terms of an A.P is given and from the sum we calculated the value of a and d using the nth term formula. . Similarly learn about geometric progression and harmonic progression. This will help a lot to solve problems related to A.P, G.P and H.P.